Closed-Loop Corrections to the S${\mathrm{U}}_3$ × S${\mathrm{U}}_3$ $\sigma$ Model: One- and Two-Point Functions

We present the calculation of one-loop corrections to the one- and two-point functions in the renormalizable S${\mathrm{U}}_3$ $\sigma$ model with a symmetry-breaking term $L_{\mathrm{SB}}={\epsilon }_0{\sigma }_0+{\epsilon }_8{\sigma }_8$. We renormalized at the masses of $\pi$, $K$, $\eta$, ${\eta }^{\prime }$, $\sigma$, and $f_{\pi }$. The second-order corrections are found to be small compared to tree-approximation values. The measure of octet breaking, $b=\frac{〈{\sigma }_8〉}{\sqrt{2}〈{\sigma }_0〉}$, changes less than 5%. The value of $a=\frac{{\epsilon }_8}{\sqrt{2}{\epsilon }_0}$ is insignificantly changed. The scalar-meson masses are shifted by less than 10%, with the exception of the ${\sigma }^{\prime }$. The widths are large, with the exception of the ${\pi }_N$. We calculate corrections to $\frac{f_K}{f_{\pi }}$, the wave-function renormalization constants, mixing angles, and the renormalized $\pi$ and $K$ propagators at $q^2=0\mathrm{}$.